New light on solving the sextic by iteration:
An algorithm using reliable dynamics

In recent work on holomorphic maps that are symmetric under
certain complex
reflection groups---generated by complex reflections through a set of
hyperplanes, the
author announced a general conjecture related to reflection groups. The
claim is that
for each reflection group $G$, there is a $G$-equivariant holomorphic map
that is
critical exactly on the set of reflecting hyperplanes.
One such group is the Valentiner action $\mathcal{V}$---isomorphic to the
alternating
group $\mathcal{A}_6$---on the complex projective plane. A previous
algorithm that
solved sixth-degree equations harnessed the dynamics of a
$\mathcal{V}$-equivariant.
However, important global dynamical properties of this map were unproven.
Revisiting
the question in light of the reflection group conjecture led to the
discovery of a
degree-31 map that is critical on the 45 lines of reflection for
$\mathcal{V}$.
The map's critical finiteness provides a means of proving its
possession of the
previous elusive global properties. Finally, a sextic-solving procedure
that employs
this map's reliable dynamics is developed.